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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 24716, 8 pages doi:10.1155/2007/24716 Research Article A Multidimensional Functional Equation Having Quadratic Forms as Solutions Won-Gil Park and Jae-Hyeong Bae Received 7 July 2007; Accepted 3 September 2007 Recommended by Vijay Gupta We obtain the general solution and the stability of the m-variable quadratic functional equation f (x 1 + y 1 , ,x m + y m )+ f (x 1 − y 1 , ,x m − y m ) = 2 f (x 1 , ,x m )+2f (y 1 , , y m ). The quadratic form f (x 1 , ,x m ) = 1≤i≤ j≤m a ij x i x j is a solution of the given func- tional equation. Copyright © 2007 W G. Park and J H. Bae. This is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, dist ribu- tion, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, let X and Y be real vector spaces. A mapping f is called a quadratic form if there exist a ij ∈ R (1 ≤ i ≤ j ≤ m)suchthat f x 1 , ,x m = 1≤i≤ j≤m a ij x i x j (1.1) for all x 1 , ,x m ∈ X. For a mapping f : X m →Y, consider the m-variable quadratic functional equation f x 1 + y 1 , ,x m + y m + f x 1 − y 1 , ,x m − y m = 2 f x 1 , ,x m +2f y 1 , , y m . (1.2) When X = Y = R ,thequadraticform f : R m →R given by f x 1 , ,x m = 1≤i≤ j≤m a ij x i x j (1.3) is a solution of (1.2). 2 Journal of Inequalities and Applications For a mapping g : X →Y, consider the quadratic functional equation g(x + y)+g(x − y) = 2g(x)+2g(y). (1.4) In 1989, Acz ´ el [1] proposed the solution of (1.4). Later, many different quadratic func- tional equations were solved by numerous authors [2–6]. In this paper, we investigate the relation between (1.2)and(1.4). And we find out the general solution and the generalized Hyers-Ulam stability of (1.2). 2. Results The m-variable quadratic functional equation (1.2) induces the quadratic functional equation (1.4)asfollows. Theorem 2.1. Let f : X m →Y be a mapping satisfying (1.2)andletg : X→Y be the mapping given by g(x): = f (x, ,x) (2.1) for all x ∈ X, then g satisfies (1.4). Proof. By (1.2)and(2.1), g(x + y)+g(x − y) = f (x + y, ,x + y)+ f (x − y, ,x − y) = 2 f (x, ,x)+2f (y, , y) = 2g(x)+2g(y) (2.2) for all x, y ∈ X. The quadratic functional equation (1.4) induces the m-variable quadratic functional equation (1.2) with a n additional condition. Theorem 2.2. Let a ij ∈ R (1 ≤ i ≤ j ≤ m) and g : X→Y be a mapping satisfying (1.4). If f : X m →Y is the mapping given by f x 1 , ,x m := m i=1 a ii g x i + 1 4 1≤i<j≤m a ij g x i + x j − g x i − x j (2.3) for all x 1 , ,x m ∈ X, then f satisfies (1.2). Furthermore, (2.1)holdsif 1≤i≤ j≤m a ij = 1. (2.4) W G. Park and J H. Bae 3 Proof. By (1.4)and(2.3), f x 1 + y 1 , ,x m + y m + f x 1 − y 1 , ,x m − y m = m i=1 a ii g x i + y i + g x i − y i + 1 4 1≤i<j≤m a ij g x i + y i + x j + y j − g x i + y i − x j − y j + 1 4 1≤i<j≤m a ij g x i − y i + x j − y j − g x i − y i − x j + y j = 2 m i=1 a ii g x i + g y i + 1 4 1≤i<j≤m a ij g x i + y i + x j + y j + g x i − y i + x j − y j − 1 4 1≤i<j≤m a ij g x i + y i − x j − y j + g x i − y i − x j + y j = 2 m i=1 a ii g x i + g y i + 1 2 1≤i<j≤m a ij g x i + x j + g y i + y j − g x i − x j − g y i − y j = 2 f x 1 , ,x m +2f y 1 , , y m (2.5) for all x 1 , ,x m , y 1 , , y m ∈ X. Letting x = y = 0andy = x in (1.4), respectively, g(0) = 0, g(2x) = 4g(x) (2.6) for all x ∈ X.By(2.3) and the above two equalities, f (x, ,x) = m i=1 a ii g(x)+ 1 4 1≤i<j≤m a ij g(2x) − g(0) = 1≤i≤ j≤m a ij g(x) = g(x) (2.7) for all x ∈ X. Example 2.3. The function g : R→R given by g(x) = x 2 satisfies (1.4). By Theorem 2.2 , the quadratic form f : R m →R given by f x 1 , ,x m = 1≤i≤ j≤m a ij x i y j (2.8) satisfies (1.2). 4 Journal of Inequalities and Applications Example 2.4. Let g : C→C be the function given by g(z) = zz. Then, it satisfies the qua- dratic functional equation (1.4). If f : C m → C is the mapping given by (2.3), that is, f z 1 , ,z m = m i=1 z i z i i j=1 a ji + 1 2 m j=i+1 a ij , (2.9) then f satisfies the m-variable quadratic functional equation (1.2). Example 2.5. Let M 2 (R) be the real vector space of all 2×2 real matrices and g : M 2 (R)→R the determinant function given by g(A) = det(A) (2.10) for all A ∈ M 2 (R). Then, it satisfies (1.4). Using (2.3), f : M 2 (R) × M 2 (R)→R is given by f (A, B) = (a 11 +(1/2)a 12 )det(A)+(a 22 +(1/2)a 12 )det(B)(a 11 ,a 12 ,a 21 ,a 22 ∈ R ). Also, f satisfies (1.2). In the following theorem, we find out the general solution of the m-variable quadratic functional equation (1.2). Theorem 2.6. A mapping f : X m →Y satisfies (1.2)ifandonlyifthereexistsymmetric biadditive mappings S 1 , ,S m : X 2 →Y and biadditive mappings M ij : X 2 →Y (1 ≤ i< j≤ m) such that f x 1 , ,x m = m i=1 S i x i ,x i + 1≤i<j≤m M ij x i ,x j (2.11) for all x 1 , ,x m ∈ X. Proof. We first assume that f is a solution of (1.2). Define f 1 , , f m : X→Y by f 1 (x):= f (x,0, ,0), , f m (x):= f (0, ,0,x)forallx ∈ X. One can easily verify that f 1 , , f m are quadratic. By [1], there exist symmetric biadditive mappings S 1 , ,S m : X 2 →Y such that f 1 (x) = S 1 (x, x), , f m (x) = S m (x, x)forallx ∈ X.DefineM ij : X 2 →Y by M ij (x, y):= f (0, ,0,x,0, ,0,y,0, ,0)− f (0, ,0,x,0, ,0,0,0, ,0) − f (0, ,0,0,0, ,0,y,0, ,0) (2.12) for all i, j with 1 ≤ i<j≤ m and all x, y ∈ X. On the rig ht-hand side of (2.12), x and y are the ith and the jth components, respectively. Then, M ij are biadditive for all i, j with W G. Park and J H. Bae 5 1 ≤ i< j≤ m. Indeed, by (1.2)and(2.12), we obtain M ij x 1 + x 2 , y = f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 − f 0, ,0,x 1 + x 2 ,0, ,0,0,0, ,0 − f (0, ,0,0,0, ,0, y,0, ,0) = f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 − 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 + f 0, ,0,x 1 + x 2 ,0, ,0,−y,0, ,0 = 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 − f 0, ,0,x 1 + x 2 ,0, ,0,−y,0, ,0 = f (0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0) − 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 + f 0, ,0,x 1 + x 2 ,0, ,0,−y,0, ,0 = f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 − f 0, ,0,x 1 + x 2 ,0, ,0,0,0, ,0 − f (0, ,0,0,0, ,0, y,0, ,0) = 1 2 2 f 0, ,0,x 1 + x 2 ,0, ,0,y,0, ,0 +2f (0, ,0,0,0, ,0,y,0, ,0) − 2f 0, ,0,x 1 + x 2 ,0, ,0,0,0, ,0 − 2 f (0, ,0,0,0, ,0,y,0, ,0) = 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,2y,0, ,0 − f 0, ,0,x 1 + x 2 ,0, ,0,0,0, ,0 − 2 f (0, ,0,0,0, ,0,y,0, ,0) = 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,2y,0, ,0 + f 0, ,0,x 1 − x 2 ,0, ,0,0,0, ,0 − 1 2 f 0, ,0,x 1 + x 2 ,0, ,0,0,0, ,0 + f 0, ,0,x 1 − x 2 ,0, ,0,0,0, ,0 − 2 f (0, ,0,0,0, ,0,y,0, ,0) = f 0, ,0,x 1 ,0, ,0,y,0, ,0 + f 0, ,0,x 2 ,0, ,0,y,0, ,0 − f 0, ,0,x 1 ,0, ,0,0,0, ,0 − f 0, ,0,x 2 ,0, ,0,0,0, ,0 − 2 f (0, ,0,0,0, ,0,y,0, ,0) = f 0, ,0,x 1 ,0, ,0,y,0, ,0 − f 0, ,0,x 1 ,0, ,0,0,0, ,0 + f (0, ,0,0,0, ,0,y,0, ,0) + f 0, ,0,x 2 ,0, ,0,y,0, ,0 − f 0, ,0,x 2 ,0, ,0,0,0, ,0 + f (0, ,0,0,0, ,0,y,0, ,0 = M ij 0, ,0,x 1 ,0, ,0,y,0, ,0 + M ij 0, ,0,x 2 ,0, ,0,y,0, ,0 (2.13) 6 Journal of Inequalities and Applications for all x 1 ,x 2 , y ∈ X. Similarly, M ij 0, ,0,x,0, ,0,y 1 + y 2 ,0, ,0 = M ij 0, ,0,x,0, ,0,y 1 ,0, ,0 + M ij 0, ,0,x,0, ,0,y 2 ,0, ,0 (2.14) for all x, y 1 , y 2 ∈ X. Conversely, we assume that there exist symmetric biadditive mappings S 1 , ,S m :X 2 →Y and biadditive mappings M ij : X 2 →Y (1 ≤ i< j≤ m)suchthat f x 1 , ,x m = m i=1 S i x i ,x i + 1≤i<j≤m M ij x i ,x j (2.15) for all x 1 , ,x m ∈ X.SinceM ij (1 ≤ i< j≤ m) are biadditive and S 1 , ,S m are symmetric biadditive, f x 1 + y 1 , ,x m + y m + f x 1 − y 1 , ,x m − y m = m i=1 S i x i + y i ,x i + y i + 1≤i<j≤m M ij x i + y i ,x j + y j + m i=1 S i x i − y i ,x i − y i + 1≤i<j≤m M ij x i − y i ,x j − y j = m i=1 S i x i ,x i +2S i x i , y i )+S i y i , y i + 1≤i<j≤m M ij x i ,x j + M ij x i , y j + M ij y i ,x j + M ij y i , y j + m i=1 S i x i ,x i − 2S i x i , y i + S i y i , y i + 1≤i<j≤m M ij x i ,x j − M ij x i , y j − M ij y i ,x j + M ij y i , y j = 2 m i=1 S i x i ,x i + 1≤i<j≤m M ij x i ,x j +2 m i=1 S i y i , y i + 1≤i<j≤m M ij y i , y j = 2 f x 1 , ,x m +2f y 1 , , y m (2.16) for all x 1 , ,x m , y 1 , , y m ∈ X. Let Y be complete and let ϕ : X 2m →[0,∞) be a function satisfying ϕ x 1 , ,x m , y 1 , , y m := ∞ j=0 1 4 j+1 ϕ 2 j x 1 , ,2 j x m ,2 j y 1 , ,2 j y m < ∞ (2.17) for all x 1 , ,x m , y 1 , , y m ∈ X. W G. Park and J H. Bae 7 Theorem 2.7. Let f : X m →Y be a mapping such that f x 1 + y 1 , ,x m + y m + f x 1 − y 1 , ,x m − y m − 2 f x 1 , ,x m − 2 f y 1 , , y m ≤ ϕ x 1 , ,x m , y 1 , , y m (2.18) for all x 1 , ,x m , y 1 , , y m ∈ X. Then, there exists a unique m-variable quadratic mapping F : X m →Y such that f x 1 , ,x m − F x 1 , ,x m ≤ ϕ x 1 , ,x m ,x 1 , ,x m (2.19) for all x 1 , ,x m ∈ X. The mapping F is given by F x 1 , ,x m := lim j→∞ 1 4 j f 2 j x 1 , ,2 j x m (2.20) for all x 1 , ,x m ∈ X. Proof. Letting y 1 = x 1 , , y m = x m in (2.18), we have f x 1 , ,x m − 1 4 f (0, ,0)+ f 2x 1 , ,2x m ≤ 1 4 ϕ x 1 , ,x m ,x 1 , ,x m (2.21) for all x 1 , ,x m ∈ X. T hus, we obtain 1 4 j f 2 j x 1 , ,2 j x m − 1 4 j+1 f (0, ,0)+ f 2 j+1 x 1 , ,2 j+1 x m ≤ 1 4 j+1 ϕ 2 j x 1 , ,2 j x m ,2 j x 1 , ,2 j x m (2.22) for all x 1 , ,x m ∈ X and all j.Forgivenintegersl, n (0 ≤ l<n), we get 1 4 l f 2 l x 1 , ,2 l x m − 1 4 n f (0, ,0)+ f 2 n x 1 , ,2 n x m n−1 j=l 1 4 j+1 ϕ 2 j x 1 , ,2 j x m ,2 j x 1 , ,2 j x m (2.23) for all x 1 , ,x m ∈ X.By(2.23), the sequence {(1/4 j ) f (2 j x 1 , ,2 j x m )} is a Cauchy se- quence for all x 1 , ,x m ∈ X.SinceY is complete, the sequence {(1/4 j ) f (2 j x 1 , ,2 j x m )} converges for all x 1 , ,x m ∈ X.DefineF : X m →Y by F x 1 , ,x m := lim j→∞ 1 4 j f 2 j x 1 , ,2 j x m (2.24) 8 Journal of Inequalities and Applications for all x 1 , ,x m ∈ X.By(2.18), we have 1 4 j f 2 j x 1 + y 1 , ,2 j x m + y m + 1 4 j f 2 j x 1 − y 1 , ,2 j x m − y m − 2 4 j f 2 j x 1 , ,2 j x m − 2 4 j f 2 j y 1 , ,2 j y m ≤ 1 4 j ϕ 2 j x 1 , ,2 j x m ,2 j y 1 , ,2 j y m (2.25) for all x 1 , ,x m , y 1 , , y m ∈ X and all j. Letting j→∞ and using (2.17), we see that F sat- isfies (1.2). Setting l = 0 and taking n→∞ in (2.23), one can obtain the inequality (2.19). If G : X m →Y is another m-variable quadratic mapping satisfying (2.19), we obtain F x 1 , ,x m − G x 1 , ,x m = 1 4 n F 2 n x 1 , ,2 n x m − G 2 n x 1 , ,2 n x m ≤ 1 4 n F 2 n x 1 , ,2 n x m − f 2 n x 1 , ,2 n x m + 1 4 n f 2 n x 1 , ,2 n x m − G 2 n x 1 , ,2 n x m ≤ 2 4 n ϕ 2 n x 1 , ,2 n x m ,2 n x 1 , ,2 n x m −→ 0asn −→ ∞ (2.26) for all x 1 , ,x m ∈ X. Hence, the mapping F is the unique m-variable quadratic mapping, as desired. References [1] J. Acz ´ el and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of MathematicsandItsApplications, Cambridge University Press, Cambridge, UK, 1989. [2] J H. Bae and K W. Jun, “On the generalized Hyers-Ulam-Rassias stability of an n-dimensional quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 183–193, 2001. [3] J H. Bae and W G. Park, “On the generalized Hyers-Ulam-Rassias stability in Banach modules over a C ∗ -algebra,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 196– 205, 2004. [4] J H. Bae and W G. Park, “On stability of a functional equation with n-variables,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 4, pp. 856–868, 2006. [5] S M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126–137, 1998. [6] W G. Park and J H. Bae, “On a bi-quadratic functional equation and its stability,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 4, pp. 643–654, 2005. Won-Gil Park: National Institute for Mathematical Sciences, 385-16 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea Email address: wgpark@nims.re.kr Jae-Hyeong Bae: Department of Applied Mathematics, Kyung Hee University, Yongin 449-701, South Korea Email address: jhbae@khu.ac.kr . functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126–137, 1998. [6] W G. Park and J H. Bae, “On a bi -quadratic functional. Hyers-Ulam-Rassias stability in Banach modules over a C ∗ -algebra,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 196– 205, 2004. [4] J H. Bae and W G. Park, “On stability. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 24716, 8 pages doi:10.1155/2007/24716 Research Article A Multidimensional Functional Equation Having Quadratic
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